A factorization of the form

(1)

The factorization for was discovered by Aurifeuille, and the general form was
subsequently discovered by Lucas. The large factors are sometimes written as
and
as follows:

which can be written

where
and

See also Gauss's Cyclotomic Formula
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References Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations
of b -n +/-1, b =2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, rev.
ed. Providence, RI: Amer. Math. Soc., pp. lxviii-lxxii, 1988. Riesel,
H. "Aurifeullian Factorization" in Appendix 6. Prime
Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser,
pp. 309-315, 1994. Wagstaff, S. S. Jr. "Aurifeullian Factorizations
and the Period of the Bell Numbers Modulo a Prime." Math. Comput. 65 ,
383-391, 1996. Referenced on Wolfram|Alpha Aurifeuillean Factorization
Cite this as:
Weisstein, Eric W. "Aurifeuillean Factorization."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/AurifeuilleanFactorization.html

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